Lattice/Talk

The first definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not have used that name. -- JanHidders

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The definition is given as:

• A least upperbound of V is an element x in L such that
  • for all y in V it holds that y <= x, and
  • for all z in L it holds that if z <= v for all v in V then x <= z.
• A greatest lowerbound of V is an element x in S such that
  • for all y in V it holds that x <= y, and
  • for all z in L it holds that if v <= z for all v in V then z <= x.

Isn't the first inequality in the second subbullet under both of the main bullets backwards? Shouldn't it be v <= z in the first case and z <= v in the second case, rather than vice versa?

Yup.

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Aren't finitely generated subgroups of Rⁿ or Cⁿ also called lattices? I wonder if they are related to the order-lattices. --AxelBoldt

Discrete subgroups, rather than finitely-generated subgroups, I think. E.g., <1,π> is a finitely generated subgroup of R, but it isn't a lattice. They aren't related to the type of lattice described in the current article. I was going to add a mention of them yesterday, but I couldn't think of anything much to write.
Zundark, 2001-08-20

I see. Maybe Minkowski's theorem about the number of lattice points in a convex set could be linked. --AxelBoldt The new material science definition seems to be the same as a discrete subgroup. --AxelBoldt

Yes. I think what we should do is to add the discrete subgroup definition, and then modify the materials science definition to mention that this is a special case of one of the mathematical definitions. --Zundark, 2001-08-21